Book I

Newton establishes that if quantities are proportional to their differences, they must be in continued proportion. This is a foundational lemma for understanding geometric progressions of velocities under resistance.

When a body moves through a uniform resisting medium where resistance is proportional to velocity, velocities at the beginning of equal time intervals form a geometric progression, and spaces described are proportional to velocities.

The motion under velocity-proportional resistance can be represented using a rectangular hyperbola with the time as area and velocity as reciprocal function, providing a geometric tool for analysis.

Newton determines the motion of a body ascending or descending under uniform gravity and resistance proportional to velocity, showing that maximum velocity is equal to the gravitational acceleration divided by the resistance coefficient.

The maximum velocity a falling body can acquire is to the velocity acquired in any given time as the constant gravitational force is to the excess of gravity over resistance at that time.

Newton determines the curved path of a projectile under uniform gravity and resistance proportional to velocity, expressing it through the tangent construction and showing how to find velocities at any point.

Newton establishes the fundamental rule of his fluxional calculus: the moment of a generated quantity equals the sum of the moments of its component terms, each multiplied by appropriate powers and coefficients.

Newton notes his correspondence with Leibniz regarding methods for determining maxima, minima, and tangents, indicating both developed similar infinitesimal methods independently.

For a body ascending or descending under gravity with resistance, when equal spaces are traversed, the absolute forces (combining gravity and resistance) form a geometric progression.

The time of ascent is proportional to a circular sector, and the time of descent to a hyperbolic sector, with appropriate velocity representations as tangent lengths.

Given a curve and gravity field, Newton shows how to find the medium density at each point that will make a projectile follow that prescribed path.

When resistance is proportional to the square of velocity, if times form a geometric progression, velocities form the inverse geometric progression, and equal spaces are described in corresponding times.

Motion under squared-velocity resistance can be represented using a rectangular hyperbola where velocity is represented by reciprocal distances from the asymptote.

For spheres of equal density moving with equal velocities, the distance each travels while losing half its velocity is proportional to its diameter.

Bodies losing equal proportions of motion in times proportional to their initial motion divided by resistance, describing spaces proportional to initial velocities.

When resistance combines both velocity-proportional and squared-velocity components, the reciprocals of velocities form a geometric progression with appropriate adjustments.

Under mixed resistance, equal space intervals produce velocities in geometric progression that maintain constant ratio over time.

For vertical motion under gravity with mixed resistance, sectors of circles and hyperbolas represent time intervals, with velocity related to segment lengths.

The space described in mixed resistance motion relates to the difference between areas representing total force and adjusted force progression.

A body can move in a spiral intersecting all radii at equal angles in a medium whose density is reciprocal to distance from center, with centripetal force reciprocal to squared distance.

In the spiral motion of the previous proposition, velocity at any distance equals that in an unresisting circular orbit at that distance, and medium density varies inversely with the tangent angle.

A body can move in a spiral intersecting radii at constant angles in media whose density follows any power law, provided the centripetal force follows the corresponding power law.

Newton clarifies that his treatment assumes resistance is proportional to density and applies only to bodies small enough that density variations across them are negligible.